Appendix. Mathematical Theorems

Transcription

1 Appendix Mathematical Theorems This appendix introduces some theorems required in the discussion in Chap. 5 of the asymptotic behavior of minimax values in Pb-game models. Throughout this appendix, b is an integer greater than 1. Theorem 1. In the interval (0, 1) there is exactly one root of the equation (A) This root is denoted by W b Proof. Consider the function f(x) = x b + x -1. Since f is strictly monotonically increasing in the interval [0,1], and sincef(o) = -1 andf(l}= 1, there is exactly one zero in (0, I). 0 Theorem 2. The sequence { W b } with all possible b's is a strictly increasing sequence and limb"" 00 Wb= 1. Proof. Note that 0< Wb< 1 and that Wb= 1- Wbb for each b> 1. Let b<c. The assertion Wb;;::: We would lead to the following contradiction: Wb= 1-(Wb)b:s; 1-(Wl< 1-(WeY= We' Therefore, {Wb} is strictly increasing. Let limb... OOWb= W. Then Wb < W:s; 1 for all b. If W < 1, then Wbb < W b for all band limb... OOWbb=O. This leads to another contradiction: limb"" 00 Wb=limb... oo(1- Wbb) = 1-0= 1. o Theorem 3. Wb is also a root of the equation: (B) Proof. From Wbb+ Wb-1 =0, we have 1- Wb b = Wb and 1-(1- Wbb)b= 1- Wb b = Wb. o

2 Theorem 4. Wb is the only root of the equation (B) in the interval (0, 1). Appendix 101 Proof Consider the function y = 1-(1-Xb)b - x. Then y has 3 different zeros, 0, Wb, and 1. If there were another zero in (0, 1) then the derivative of y' would have at least three zeros, and y" would have at least two zeros in (0, 1). However, since y" =b2(b-1)xb- 2(1-Xb)b-2(1_(b+ 1)xb), y" has only one zero in (0, 1). Therefore, y has only one zero in (0, 1). 0 Theorem 5. The following two inequalities hold: and 1-(1-Xb)b<X for O<x< Wb Proof Let Since y' = b2xb- 1(1_ Xb)b-l -1, y'(o) and y'(1) are negative, and thus y is decreasing at 0 and 1. The asserted inequalities now come from the fact that y is zero only at 0, Wb and 1. 0 Let p be any value in the interval [0, 1]. For any integer b> 1, define P n recursively as follows: Po=P, Pn= 1-(1-Pn-/t for Theorem 6. If P = Wb, then P n = Wb for all n; otherwise, and limn-+oopn=o for O::;p< Wb, limn-+ oopn= 1 for Wb<p::; 1. n>o. Proof The first assertion in this theorem comes directly from Theorem 3. Consider next the second assertion. For the cases when p=o or 1, the conclusions come immediately from the definition. Now suppose O<p< Wb. Consider the function Then f(x) = 1-(1-Xb)b. and each P n is positive. From Theorem 5, P n is strictly decreasing. Let W=limn-+ooPn Then 0::; W < Wb. W is a fixed point off, and from Theorem 4 we have W = o. The assertion for the other case (i.e., Wb<p< 1) can be proved similarly. 0

3 References Barr A, Feigenbaum EA (eds) (1981) The handbook of artificial inteliigence, vol!. WilIiam Kauffmann, Los Altos, CA Baudet GM (1978) On the branching factor of the alpha-beta pruning algorithm. Artificial IntelIigence 10(2): Beal D (1980) An analysis of minimax. In: Clarke MRB (ed) Advances in computer chess 2. Edinburgh University Press, Edinburgh, pp Beal D (1982) Benefits of minimax search. In: Clarke MRB (ed) Advances in computer chess 3. Pergamon, Oxford, pp Bratko I, Gams M (1982) Error analysis of the minimax principle. In: Clarke MRB (ed) Advances in computer chess 3. Pergamon, Oxford, pp 1-15 Chung KL (1974) A course in probability theory, 2nd ed Academic Press, New York Dresher M (1981) The mathematics of games of strategy: theory and application. Dover Publications, New York Knuth DE, Moore RW (1975) An analysis of alpha-beta pruning. Artificial IntelIigence 6(4): Nau DS (1980) Decision quality as a function of search depth of game trees. TR -866, Computer Science Department, University of Maryland Nau DS (1981) An investigation of the causes of pathology in games. TR-999, Computer Science Department, University of Maryland Nau DS (1983a) Pathology on game trees revisited, and an alternative to minimaxing. Artificial IntelIigence 21: Nau DS (1983b) Game graph structure and its influence on pathology. TR-1246, Computer Science Department, University of Maryland Nau DS, Purdom P, Tzeng C-H (1986) An evaluation of two alternatives to minimax. In: Kanaland LN, Lemmer JE (eds) Uncertainty in artificial inteliigence. North-HolJand, Amsterdam, pp Nilsson N (1980) Principles of artificial intelligence. Tioga, Palo Alto, CA Pearl J (1980) Asymptotic properties of minimal trees and game-searching procedures. Artificial IntelIigence 14: Pearl J (1981) Heuristic search theory: a survey of recent results. In: Proc IJCAI 7, Vancouver, British Columbia, Canada, pp Pearl J (1983) On the nature of pathology in game searching. Artificial IntelIigence 20: Pearl J (1984) Heuristics: inteliigent search strategies for computer problem solving. Addison-Wesley, Reading, MA Pohl I (1970) First results on the effect of error in heuristic search. In: Meltzer B, Michie D (eds) Machine inteliigence 5. American Elsevier, New York, pp Reibman AL, BaIJard BW (1983a) Non-minimax search strategies for use against faliible opponents. In: Proceedings of the national conference on artificial inteliigence AAAI-83. William Kaufmann, Los Altos, CA, pp Reibman AL, BaIJard BW (1983b) The performance of a non-minimax search strategy in games with imperfect players. CS , Duck University, Durham, NC Samuel AL (1959) Some studies in machine learning using the game of checkers. IBM Journal R&D 3: Slagle R, Dixon J (1970) Experiments with M & N tree-searching programs. Communication of ACM 13:

4 References 103 Tzeng C-H (1984) A mathematical model of heuristic game playing. In: Laubsch J (ed) GWAI-84. Springer-Verlag, Berlin, Heidelberg pp Tzeng C-H, Purdom P (1983) A theory of game trees. In: Proceedings of the national conferences on artificial intelligence AAAI-83. William Kaufmann, Los Altos, CA, pp Tzeng C-H, Purdom P (1986) Estimation of minimax values. In: Ras ZW, Zemankova M (eds) Proceedings SIGART international symposium on methodologies for intelligent systems. ACM, New York, pp von Neumann J, Morgenstern M (1946) Theory of games and economic behavior. Princeton University Press, Princeton, NJ

5 Subject Index Additivity 32 countable 32 finite 32 Adversary 2 Almost everywhere 38 Alpha-beta procedure 13, 14, 16,20, 59 AND 48, 49, 50 Asymptotic behavior 46 Average 34 Average propagation 26 Back-up process 5, 18, 20 Backgammon 7 Ballard 25, 26 Barr 21 Baudet 17 Baye's rule 37 Bayesian statistics 37 Bea1 4, 22, 98 Binomial distribution 82 BLACK 10 Bonus function 25 Boolean operators 48 Borel field 29, 30, 65 product 35 total 29 trivial 29, 38 Borel set linear 33 Branching factor 45, 81 Bratko 98 Bridge 7 Checker 98 Chess 7,98 Chung 28,34 City-block distance 3, 4 Combinatorial explosion 3 Conditional expectation 37, 38 Conditional probability 36, 82, 91, 93 Control strategy 3 Control system 1 Countable additivity 32 Cutoff alpha 15 beta 15 lower 15 upper 15 Decision behavior 40 Decision making 4, 64, 68 Decision model 68, 69 Decision problem 68 Decision quality 68, 70, 71, 72 Decision strategy 68 De Morgan's law 29 Discrete 32 Distance city-block 3 Manhattan 3 Dixon 24,25 Dresher 13, 41 8-Puzzle 1,2,4 Equation 54 system 54 Estimation 5 Estimator 65 A- 65 B- 67 more precise 67 Evaluation function static 3,4,18, 19,41,52,58 Event 29,58 Everywhere 38 Expectation 33 conditional 37, 38, 65 Face-value principle 20, 22, 69 Fallible 25 Feigenbaum 21 Finite intersection 29 Forced loss 78, 79 Forced win 22, 66, 78, 79, 82, 91, 94, 95 Fubini's Theorem 77 Function measurable 33

6 Index 105 Gambling 39 Game finite 5 G 1-10,20,47,56 G 2-47 G d- 5, 10, 20, 47 P 2-11, 20, 52 P b- 5,20,22 payoff 13 perfect information 5, 7, 18 player 18 T- 41 two-person 5, 18 WIN-LOSS 5, 66, 73, 78 zero-sum 5, 18 Game graph 9, 28, 40, 67 G 1-11,48 G 2-47 G d- 47 height 9, 40, 48, 49 Game model product 73, 74 G 1-52, 56, 88 G d- 46, 47, 87, 91, 99 P 2-52 P b- 45,81,98 Game set 54, 56 Game space P 2-28 Game theory 45 Game tree 8, 28, 40 level 10,41 link 8 nodes 9 non-terminal node 8, 9 predecessor 9 root 8,9,40 sons 8 subtree 9 successor 8, 9 terminal node 9, 40 Game value 13, 41, 42, 44, 64 Game-tree search 4, 15,52,59 heuristic 52, 59 pathology 4, 18,21,22,52,69,98 Gams 98 Global database Goal 2 Graph product 73, 74 search 2,4 Height 9,40,45,48,49,81 Heuristic information 3,4, 5, 28, 51, 57, 58, 94, 98 cumulative 52, 53, 57, 60, 62 local 75,78 product 75, 76, 77, 80 Heuristic search 3, 4, 5, 55, 59, 61, 65, 68 information 60 local 76 method 3 product 75, 76 HORIZONTAL 11 I.i.d. 45, 81 Inclusion 61 Independence 34, 73, 75 Indicator 66 Information complete 53, 59 cumulative 53, 62 partial 71 trivial 59 Integrable 34 Integration 33 Kalah 25 Knuth 17 Last player 48, 88 Leaves 40 LEFT 8 Localization 75 LOSS 23 Manhattan distance 3 Martingale 5, 28, 36, 39, 67 MAX 9, 10, 40, 43, 44, 45, 46, 47, 64 Mean 33, 34, 45 Measurable set 31, 58 Measurable function 33 Measure 31 product 35 MIN 9, 10,40, 43, 44, 45, 46, 47, 64 Minimal cost 3 Minimax procedure 13, 14,20,59 Minimax value 5, 13, 18, 40, 44, 81, 87, 89, 94 Misplaced tile 3 Moore 17 Morgenstern 13 Move chance 7 minimax optimal 14 optimal 13 personal 7

7 106 Index Nau 4, 1 I, 20, 22, 26, 27, 86, 98, 99 N -decision random variable 70, 71 N-decision-making 70, 71 Negmax 17 Nilsson 3, 19 Node LOSS 23 MAX 9, 40, 44, 49, 50, 80, 95 MIN 9,40,44,49,50,80,94 search-tip 3, 19 terminal 40 WIN 23 Node strength 28,64 Non-adversary 2 Non-comparable 61 NP-complete 5 One-counter 5, 52, 83, 88, 92 Optimal solution 3 OR 48,49,50 Partition 29 finer 30 proper refinement 30 properly finer 30 refinement 30 Pathological phenomenon 4, 18, 21, 22, 52, 69, 98 Payoff 9 final 9 Pearl 3, 4, 17, 22, 24, 45, 73, 98 Performance quality 22 Playing path 42 Pohl 4 Poker 7 PP rules 80 PP-l 79 PP-2 79 Precise equally 55, 61 more 55,61 non-comparable 61 properly more 55, 61 Predicted strength 25 Probabilistic game model 5,40,41 Probabilistic game space 32 Probability measure 30 Probability space 30 product 35 Problem graph-search 2 NP-complete 3 optimal 3 Problem state 2 Procedure *-MIN 25,26 alpha-beta 13, 14, 16,20,59 M&N 24 minimax 13, 14,20,59 product-propagation 5, 23, 24, 73, 77, 80, 81, 83,98 Product subgraph 73, 74 game model 73, 74 component 74,75 Product measure 35 Product-propagation 5,23,24,73,77,80,81,83, 98 Product set 35 Product space 35 Production rules 1, 2 Production system 1,2 Purdom 26, 27, 66 Random function 58 Random variable 33, 65 estimation 65 i.i.d., 45, 81 independent 34, 73, 75 mean 34,45 Random vector 41 Recurrence relation 46, 83, 84, 85 Reibman 25, 26 RIGHT 8 Root 40 Round 42 complete 42 Samuel 18 Search depth 21 Search event 60, 61 Search information 60,61 Search node 61,90, 93 Search value 90, 93 Singleton 59 Slagle 24, 25 Static value 19 Strategy 41 heuristic 18 minimax optimal 44, 45, 64, 68 non-randomized 42 randomized 43 Strength Subfield 29 Subgame 9 Successor 9

8 Index 107 T-game 41 Tic-tac-toe 1, 2, 19 Tzeng 26, 27,41, 66 VERTICAL 11 Visibility 22, 30, 61, 63 improved 63 von Neumann 13 Weighted average 26 WHITE 10 WIN 23 Zero-counter 61